If a and b are positive even integers, and the least common

Question

If a and b are positive even integers, and the least common multiple of a and b is expressed as a*b/n, which of the following statements could be false?

A. n is a factor of both a and b 
B. (a*b)/n < ab 
C. ab is multiple of 2. 
D. (a*b)/n is a multiple of 2. 
E. n is a multiple of 4.

blueseas · Accepted Answer

This question is easier if you know that, for any positive integers a and b:

LCM(a,b) * GCD(a,b) = a*b

So  LCM(a,b) = ab/GCD(a,b)

In this question, n is just the GCD of a and b. So we know automatically that n is a factor of a and b... A) must be true. 
If a and b are both even, they are both divisible by 2, so the smallest possible value for their GCD is 2. That is, n >= 2, so B) must be true 
If a and b are even, ab is clearly even, and so is the LCM of a and b, so C) and D) must be true. 
 E) is the only one that might be false; E will only be true if both a and b are divisible by 4.
 
hence E

gsr · Answer

E may be false.
If a=4, b=2 least comm. multiple is 4.
ab/n=4 => n=2 which is not a multiple of 4

kinjiGC · Answer

Any information is provided such that a and b cannot be a multiple of 4 ( 'a' and 'b' are positive even integers - a can be 4 and b can be 16 or 8) ? I don't think it is valid question.

EMPOWERgmatRichC · Answer

Hi All,

The last poster in this thread (kinjiGC) had some concerns about whether this question was "valid" or not, but I can assure you that it is.

The question is based on Number Property rules that exist in the realm of factors and multiples. Given the restrictions in the prompt, 4 of the answers are ALWAYS TRUE....

For example, since we're told that A and B are POSITIVE EVEN INTEGERS, we know that (A)(B) = (even)(even), so the product will ALWAYS be EVEN. Thus AB is a multiple of 2 and we can eliminate Answer C (since we're asked to find something that is SOMETIMES FALSE - meaning NOT ALWAYS TRUE).

These types of "...must be true..." questions almost always involve Number Properties of some kind, so you can TEST VALUES or use Number Property Rules to get to the correct answer. You'll see a few questions in this general "format" on Test Day and a bunch of questions involving Number Properties in general (especially in DS).

GMAT assassins aren't born, they're made,
Rich

CCMBA · Answer

If a and b are positive even integers, and the least common multiple of a and b is expressed as a*b/n, which of the following statements could be false?

A. n is a factor of both a and b 
B. (a*b)/n < ab 
C. ab is multiple of 2. 
D. (a*b)/n is a multiple of 2. 
E. n is a multiple of 4.

If we divide by  n  and get a multiple of  a  and of  b , then  n  must be a common factor. Answer A must be true.

If  a  and  b  are even integers, dividing by a factor (also an integer) results in something smaller than the product  ab . Answer B must be true.

If either  a  or  b  is even, then  ab  must be even. Answer C must be true.

If both are even, the smallest either number can be is 2. Thus, the smallest common factor must be 2. Answer D must be true.

If  b  is 6 and  a  is 2,  n  would be 6, which is not 4. E can be false.

manuchadha · Answer

Using a=2, b=4. LCM (2,4) = 4. This means ab/n can be written as 2*4/2 (because LCM(=4) the least common multiple of 'a' and 'b' is expressed as ab/n, 2*4/2=4=LCM). Thus n=2 (and a=2, b=4)
A. n is a factor of both a and b - True. 2 is factor for 2 and 4
B. (a*b)/n < ab - True (4<8)
C. ab is multiple of 2. - True (8 is multiple of 2)
D. (a*b)/n is a multiple of 2. - True (4 is multiple of 2)
E. n is a multiple of 4. False (n=2 is not multiple of 4)

Shrivathsan · Answer

Great Explanation ! Thanks

!

kabirgandhi · Answer

Could anyone help explain this question? Since it is asking what "could be false", going about it by trying to take one example might not be the best strategy

aryanxx09 · Answer

Why can't it be option c? Let a=3, b=7, n=1. LCM is 21, which is 3*7/1
Can someone explain?  vikramm

WhitEngagePrep · Answer

One thing to note, is that I've never seen an official question ask "which of the following could be false, " so I'm not entirely confident on the quality of the source material. That said, the language of "could be" or "must be" is a signal for me that a question is all about Testing Cases (think process of elimination). In this case, if something COULD be false, it means that all of these answers will be true, but only one will have a situation where it might not actually hold. So we can't just test a single number - we have to test until we are able to finally break one (and show a case where it is false). When Testing Cases, I usually start with the laziest / easiest / most restrictive option I can come up with, just to see the pattern (or to test limit or edge cases). Case 1: They don't say that a and b have to be different, so let's say a = b = 2. The LCM of a and b would also be 2. Therefore, ab/n = 2 (2)(2)/n = 2 n = 2 Now test the choices: (A) 2 is a factor of both 2 and 2 - TRUE (B) 2*2/2 < 2*2 - TRUE (C) 2*2 is a multiple of 2 - TRUE (D) 2*2/2 is a multiple of 2 - TRUE (D) 2 is a multiple of 4 - FALSE We hit a false on the first try and so we can stop here! If we had hit all true, we'd have to start to be more creative The case you tested violates the initial premise that a and b are both even integers. If we look at the theory of C, if both a and b are even, then they are both multiplies of 2. Their product is actually guaranteed to be a multiple of 4, so it will definitely be a multiple of 2!

Whit

aryanxx09 · Answer

Oops! My bad. Thanks for the help.

kabirgandhi · Answer

Thank you for the explanation!

I came across this question on the free mock on the LBS website. Not sure who it is administered by - Economist/Kaplan? Seem to have questions from both.

However, good practice of number systems, could be/ must be, factors, multiples, etc.

If a and b are positive even integers, and the least common

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If a and b are positive even integers, and the least common

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